HMI & Photospheric Flows

1. HMI & Photospheric Flows • Review of methods to determine surface plasma flow; • Comparisons between methods; • Data requirements; • Necessary computational resources; • Possible improvements to methods.

2. General Approach • From 2D data arrays, f1(x1,x2) & f2(x1,x2), find vector flow v(x1,x2) consistent with: • Observed evolution, f(x1,x2) = f2(x1,x2) – f1(x1,x2) • Other possible assumptions: • Magnetic induction eqn., Bn/t = t(vnBt-vtBn) • Continuity equation, f/t + t(vtf) = 0 • Doppler velocities – more later • v(x1,x2) might have 2 or 3 components

3. General Approach, cont’d: • Ideally, with finite difference equations, cadence should beat “Courant cadence” tC = x/vmax • analog of numerical Courant condition: • time step limited by propagation speed of information • x  pixel size; vmax  expected max. flow speed • “low cadence” is t> tC • ttC very rare in solar physics! (usually, t>>tC)

4. HMI Capabilities • Pixels = .5” ~ 363 km, resolution ~ 1.5” ~ 1100 km • Photospheric csound  (kT/m)1/2 9 km/s • Courant Cadence: tHMI(363 km)/(9 km/s)  40 sec. • LOS Mag. Field Cadence, tLOS ~ 60 sec. • Vector Mag. Field: tVEC ~ 600 sec. • Typical v ~ 2 km/s, and resolution ~ 1100 km, so tPRACTICAL ~ 550 sec.

5. Current Methods • Local Correlation Tracking (LCT) • “Inductive” Methods (ILCT, MEF, …) • Feature Tracking (FT)

6. 1. Local Correlation Tracking (LCT) • Take subregions,  pixels wide, of f1 & f2, find, e.g., • shift xthat minimizes difference f ; or • shift xof peak in (Fourier) correlation func’n • Sub-pixel shifts found by interpolation – SLOW! • Most algorithms solve advection equation, f/t + (vtt) f = 0 • Can be used on intensity images, LOS, & vector magnetograms from HMI. • Cadence must be slow enough that fnoise < fadvection • Workable with very low cadence data: t 100tC

7. LCT applied to magnetograms: Démoulin & Berger’s (2003) analysis of flux transport velocity Motion of flux across photosphere, uf, is a combination of horizontal & vertical flows acting on non-vertical fields. HMI Planning Meeting

8. LCT, cont’d Hence, flows uLCT from LCT on magnetograms: • are not generally identical to plasma velocity v • solve advection equation, not continuity equation • Given vector B, can assume uf = uLCT, and thereby find v from uLCT algebraically (ADC). • Q: How good does LCT do? A: Pretty good!

9. HMI Planning Meeting

10. HMI Planning Meeting

11. A Comparable Data Set:Flare Genesis Experiment • Balloon-borne (Antarctic) observations of NOAA 8844, 25 Jan 2000 • 54 vector magnetograms, ~2.3/5.3 min. per • hi-res: .18” pixels (130.5 km), ~520 x 520 pix • LCT differenced over ti +/-10, for Dt ~85 min. • Doppler maps, too! (No info. on method.) • Tracking of white light images underway HMI Planning Meeting

12. FGE Movie HMI Planning Meeting

13. FGE: White Light vs. Mag HMI Planning Meeting

14. FGE: Larger  HMI Planning Meeting

15. Modifications to LCT • Near future: Improvement in sub-pixel interpolation – added speed. • Future: Convert to FORTRAN; parallelize. • Compute on tiles, not on each pixel. HMI Planning Meeting

16. 2. Inductive Methods • Use finite diff. approx. to magnetic induction equation’s normal comp. as add’l constraint. • Purely inductive methods need ttC • Methods currently available: ILCT, MEF, Kusano et al. (2002), MSR (Georgoulis et al., 2005, in prep.) • All methods return (vx, vy, vz) at photosphere, where (vB) = 0; parallel flow unconstrained by ind’n eqn. • Post-processing with Doppler data can give v || B • NB: NOT Doppler from Stokes I (Chae et al., 2004)

17. Directly measured Derived Inductively Inductively Derived Flows are Consistent with Induction Eqn’s Normal Component!

18. What about other components? Directly measured Derived Inductively From NLFFF Extrapolation? Derived by new method? at photosphere, z = 0 above photosphere, z > 0

19. A) ILCT: Modify LCT solution to match induction equation • Solve for , with 2D divergence and 2D curl (n-comp), and the approximation that uf=uLCT: Let NB: if only BLOS is known, we can still solve for , ! HMI Planning Meeting

20. B) Minimum Energy Fit (MEF) • Also uses induction equation’s normal component to derive flow, with additional assumption that integral of squared velocity is minimized. • Applicable to vector magnetograms. • More from D. Longcope, shortly!

21. Other Inductive Methods • Kusano et al. (2002): get v from LCT flow, derive additional flow for consistency with induction equation. • Georgoulis (2005, in prep): Use (i) “minium structure” & (ii) “coplanarity” assumptions, with (iii) induction equation to derive (iv) velocity perpendicular to magnetic field. (System overconstrained.)

22. Prelim. Comparison of Inductive Methods • Used MHD simulations of Magara (2001) • Given B(x,y,z=0,t), “practioners” computed v(x,y,z=0,t), and were then told actual v. HMI Planning Meeting

23. Some Prelim Comparisons HMI Planning Meeting

24. Some Prelim Comparisons HMI Planning Meeting

25. Some Prelim Comparisons HMI Planning Meeting

26. Some Prelim Comparisons HMI Planning Meeting

27. 3. Feature Tracking • Useful with WL images & magnetograms. • Algorithms: • White Light: L. Strous • Active region fields: B. Welsch, G. Barnes • Quiet Sun fields: C. DeForest, M. Hagenaar, C. Parnell, B. Welsch • Does not return v(x,y); rather, gives velocity of “patches” of photosphere. • Easily incorporated in pipeline.

28. Feature Tracking in AR 8038 HMI Planning Meeting

29. Conclusions • Planned data cadences are compatible with existing velocity inversion algorithms. • LCT can be used to derive flows in HMI’s intensity, LOS, and vector field maps. • ILCT, MEF suitable for determining three-component photospheric magnetic flows. • Doppler data from Stokes’ profiles (zero crossing of V, or central minima of Q,U) desirable. • Significant improvement in computational performance of LCT algorithms is needed for real-time analysis. HMI Planning Meeting